# Complexity class (P/NP) variants of Hamiltonian paths problems

I know that the following problems related to Hamiltonian paths in graph are NP-complete:

• Undirected Hamiltonian circuit:
Given an undirected graph, does it has a cycle that passes through each node exactly once?
• Undirected Hamiltonian path:
Given an undirected graph, does it has a path that passes through each node exactly once?
• Directed Hamiltonian circuit:
Given a directed graph, does it has a directed cycle that passes through each node exactly once?

But then my textbook says following problems are NP-Hard but not NP-complete:

• Problem A: Finding a Hamiltonian circuit in a graph with number of vertices $|V|$ divisible by 3
• Problem B: Determining if Hamiltonian circuit exists in the graphs as given problem A.

Why is this so?